Recently I listened to Swami Sarvapriyananda saying his "last name" is Puri, what about Swami Vivekananda? Does he belong to Adi Shankara parampara? And what about other modern swamis like Sivananda and Chinmayananda?
The last name you are talking about is the name given to an advaitan monk in Daśanāmi Saṃpradāya.
The tradition gets its name because the followers of that tradition will be given ten names as suffixes after their Sannyasa name. The ten names are:
These names are to be distributed among the four traditional Amnaya peethams established by Adi Shankaracharya. However, you will find it's nominal and sanyasis who have these names are not strictly affiliated to four peethams. It is traditionally said that Adi Shankaracharaya organized the Daśanāmi Saṃpradāya when he established four peethams in four directions.
Since, Ramakrishna Paramahamsa was initiated by a sanyasi named Totapuri, he also had a suffix puri. Even after Ramakrishna Paramahamsa, Vivekananda's disciples also got the suffix puri. Wikipedia lists many names of disciples of Ramakrishna and Vivekananda who have the suffix puri. Sivananda and Chinamaynanda have suffixes Sarasvati after their Sannyasa names.
According to Advaita-vedanta.org, (emphasis mine)
Modern Institutions: In addition to the more traditional advaita maThas and akhADas, various sannyAsIs of the daSanAmI order have established some of the more well-known modern institutions, like the Ramakrishna Math and Mission (swAmI vivekAnanda), the Self-Realization Fellowship (paramahamsa yogAnanda), the Divine Life Society (swAmI SivAnanda), Yoga Vedanta Center (swAmI vishNudevAnanda), the Chinmaya Mission (swAmI cinmayAnanda), and others. Among these, the founders of the Ramakrishna Mission, the Divine Life Society and the Chinmaya Mission trace their spiritual descent through the Sringeri paramparA. The Self-Realization Fellowship has links to the Puri paramparA.
Vivekananda and Chinmayananda do not belong strictly to Adi Shankara lineage but they trace their origin to the mathas established by him. They are from Daśanāmi Saṃpradāya.
Sri Totapuri, a great Vedanti, was one of Sri Ramakrishna's gurus. Probably that's where the sanyasis of Sri Ramakrishna Mission get their "Puri" title. You can read more about Totapuri here:
Both Sri Ramakrishna and Swami Vivekananda used to talk about Sri Shankara often. And some devotees of Swami Vivekananda think that Swamiji was a combination of Shankara's Jnana and Buddha's sacrifice.
However, I don't believe Swami Vivekananda belongs to Shankara parampara in the sense of lineage.